Tree-like Properties of Cycle Factorizations

Abstract

We provide a bijection between the set of factorizations, that is, ordered (n − 1)-tuples of transpositions in S n whose product is (12...n), and labelled trees on n vertices. We prove a refinement of a theorem of J. Dénes (1959, Publ. Math. Inst. Hungar. Acad. Sci. 4, 63-71) that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a factorization corresponds naturally under our bijection to leaf edges (incident with a vertex of degree 1) of a tree. Moreover, we give a generalization of this fact. © 2002 Elsevier Science (USA)

Generalized Non-Crossing Partitions and Buildings

“…In type A the map is surjective by Lemma 2.5 of [GY02]. But the surjectivity fails in other types, as can be seen in type B.…”

“…Using the pictorial interpretation of the σ i given in Proposition 5.8 and a characterization of all the non-crossing labeled spanning trees that correspond to a vertex in H(S n ) given in Theorem 2.2 of [GY02] (see also [AR14, Prop. 3.5.…”

Trajectories in random minimal transposition factorizations

“…, n) into transpositions (it is indeed easy to see that at least n − 1 transpositions are required to factorize a n-cycle). In the sequel, the elements of M n will be simply called minimal factorizations of size n. It is known since Dénes (1959) that |M n | = n n−2 and bijective proofs were later given by Moszkowski (1989), Goulden and Pepper (1993), Goulden and Yong (2002).…”

“…As a concrete example, the limiting Poisson distribution for # ‹ M Our result is the following distributional identity (for convenience, we state it with the objects T (n) , M (n) defined exactly as T (n) , ‹ M (n) when replacing F (n) with F (n) ). The proof is based on a bijection of Goulden and Yong (2002) between Cayley trees and minimal factorizations.…”

Permutation Factorizations and Prime Parking Functions

“…There have been several combinatorial proofs for the case λ = (n) (see [5,6,9]). Recently, Kim and Seo [7] gave a combinatorial proof of the case λ = (n) and (1, n − 1) using parking functions.…”